---
tags: 機率與統計
title: 第十五週活動
---

# 活動一
A certain pen has been designed so that true average writing lifetime under controlled conditions (involving the use of a writing machine) is ate least 10 hours. A random sample of 18 pens is selected, the writing lifetime of each is determined, and a normal quantile plot of the resulting data supports the use of a one-sample t test.

(a) What hypotheses should be tested if the investigations believe a priori that the design specification has been satisfied?
>$H_0 = 10$
>$H_1 < 10$

(b) What conclusion is appropriate if the hypotheses of part (a) are tested, t = -2.3, and α = 0.05?
>$df = 18-1$
>$t = -1.740$
>$-2.3 < -1.7$
>沒有證據拒絕$H_0$
>所以這批筆不合格

(c) What conclusion is appropriate if the hypotheses of part (a) are tested, t = -1.8, and α = 0.01?
>$df = 18-1$
>$t = -2.567$
>$-1.8 > -2.567$
>$證據顯示無法拒絕H_0$

(d) What should be concluded if the hypotheses of part (a) are test, and t = -3.6?
>$\alpha = 0.001$
>$不論 \alpha=0.01或是0.05，t值都會在基準的左邊，所以我們無法拒絕H_0$
>$所以都是不合格$
>$所以\alpha要小於0.001才有可以接受$

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# 活動二

Urban storm water can be contaminated by many sources, including discarded batteries. When ruptured, these batteries release metals of environmental significance. The article "Urban battery litter" (J. of environment engineering, 2009–pp.46-57) presented to summarize data for characteristics of a variety of batteries found in urban areas around Cleveland.

Here are data on zinc mass (g) for two different brands of size D batteries:

**Brand Sample Size Sample Mean Sample SD**
**Duracell** 15 138.52 7.76
**Energizer** 20 149.07 1.52

Assuming that both zinc mass distributions are at least approximately normal. Carry out a test at significance level .05 to decide whether true average zinc mass is different for the two types of batteries.
>$H_0 = u_1 = u_2$
>$H_1 = u_1 \ne u_2$
>$\frac{(\overline{x_1}-\overline{x_2})-(u_1-u_2)}{\sqrt{{\frac{s_1^2}{n_1}}+\frac{s_2^2}{n_2}}}=t$
>$df=14 代公式$
>$\alpha=0.05,t_{\alpha}=[2.145,-2.145]$
>$t=-5.19$
>$t<t_{\alpha}所以兩者有顯著差異$

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# 活動三
The drug diethylstilbrstrol was used for years by women as a nonsteroidal treatment for pregnancy maintenance, but it was banned in 1971 when research indicated a link with the incidence of cervical cancer. An article discussed a study in which 10 males exposed to DES and their unexposed brothers underwent various tests. This is the summary data on the results of a spatial ability test:

exposed mean = 12.6

unexposed mean = 13.8

standard error of difference = sdn√

Does DES exposure appeared to be associated with reduced spatial ability? State and test the appropriate hypotheses α =.05. Does the conclusion change if α =.01 is used?
>$H_0 = u_1-u_2 = 0$
>$H_1 = u_1-u_2 = \phi$
>$\alpha = 0.05$
>$\overline d = \overline x_1 - \overline x_2$
>$df = 9$
>$t=\frac{\overline d - \phi}{\frac{s_\alpha}{\sqrt{n}}}=\frac{(12.6-13.8)-\phi}{0.5}=-2.4$
>$t_\alpha = -1.833$
>$t<t_\alpha，有證據顯示可拒絕H_0$
>$當\alpha = 0.01，t_\alpha=2.896，無法拒絕H_0$